I was solving the following problem:
b) Show that homomorphisms from $\mathbf{Z}$ to any group $G$ is in bijetion[sic] with elements of $G$.
c) How about homomorphisms from $C_k$ (cyclic group with $k$ elements) to a group $G$? How many does there exist[sic] from $C_3$ to $S_6$? From $C_2$ to $S_6$?
I am not able to understand the question and how to proceed. Please help me in understanding the question.
Hint: For $\Bbb Z$, the homomorphisms are completely determined by where you send a generator. There are no relations among the elements ($\Bbb Z$ is a free group). Thus we get a homomorphism $h:\Bbb Z\to G$ for every choice of $h(1)\in G$.
In contrast, for $C_k$, you don't always get a nontrivial homomorphism into $G$. $C_k$ isn't free, as it is finite. As a result, the order of the image of the generator must divide $k$.
Thus there are the number of elements of order $2$ from $C_2$ to $S_6$. (I count $75$.)
And the number of elements of order $3$ from $C_3$ to $S_6$. (I count $80$.)
See this, as a double-check for the counting.